| L(s) = 1 | + 2-s − 1.27·3-s + 4-s − 0.887·5-s − 1.27·6-s + 7-s + 8-s − 1.38·9-s − 0.887·10-s − 3.41·11-s − 1.27·12-s + 1.57·13-s + 14-s + 1.12·15-s + 16-s − 1.11·17-s − 1.38·18-s + 0.542·19-s − 0.887·20-s − 1.27·21-s − 3.41·22-s − 1.27·24-s − 4.21·25-s + 1.57·26-s + 5.57·27-s + 28-s + 8.58·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.734·3-s + 0.5·4-s − 0.396·5-s − 0.519·6-s + 0.377·7-s + 0.353·8-s − 0.461·9-s − 0.280·10-s − 1.02·11-s − 0.367·12-s + 0.436·13-s + 0.267·14-s + 0.291·15-s + 0.250·16-s − 0.269·17-s − 0.326·18-s + 0.124·19-s − 0.198·20-s − 0.277·21-s − 0.727·22-s − 0.259·24-s − 0.842·25-s + 0.308·26-s + 1.07·27-s + 0.188·28-s + 1.59·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 1.27T + 3T^{2} \) |
| 5 | \( 1 + 0.887T + 5T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 - 1.57T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 - 0.542T + 19T^{2} \) |
| 29 | \( 1 - 8.58T + 29T^{2} \) |
| 31 | \( 1 - 1.49T + 31T^{2} \) |
| 37 | \( 1 - 9.01T + 37T^{2} \) |
| 41 | \( 1 - 8.13T + 41T^{2} \) |
| 43 | \( 1 + 6.86T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 - 6.27T + 53T^{2} \) |
| 59 | \( 1 - 0.503T + 59T^{2} \) |
| 61 | \( 1 + 6.22T + 61T^{2} \) |
| 67 | \( 1 + 0.989T + 67T^{2} \) |
| 71 | \( 1 + 3.88T + 71T^{2} \) |
| 73 | \( 1 + 4.79T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 - 2.42T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 1.37T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.53148518812367169476739070339, −6.62110591962564488300059189256, −6.03380076363803632259217166104, −5.44486653538318404106353659754, −4.72405544953181556043929181759, −4.20413295622050562162612035582, −3.08333701530155560523940338024, −2.52360299566093423014012907043, −1.23627389599137237567469465664, 0,
1.23627389599137237567469465664, 2.52360299566093423014012907043, 3.08333701530155560523940338024, 4.20413295622050562162612035582, 4.72405544953181556043929181759, 5.44486653538318404106353659754, 6.03380076363803632259217166104, 6.62110591962564488300059189256, 7.53148518812367169476739070339
